Model-free Data-Driven Inference

We present a model-free data-driven inference method that enables inferences on system outcomes to be derived directly from empirical data without the need for intervening modeling of any type, be it modeling of a material law or modeling of a prior distribution of material states. We specifically consider physical systems with states characterized by points in a phase space determined by the governing field equations. We assume that the system is characterized by two likelihood measures: one $mu_D$ measuring the likelihood of observing a material state in phase space; and another $mu_E$ measuring the likelihood of states satisfying the field equations, possibly under random actuation. We introduce a notion of intersection between measures which can be interpreted to quantify the likelihood of system outcomes. We provide conditions under which the intersection can be characterized as the athermal limit $mu_infty$ of entropic regularizations $mu_beta$, or thermalizations, of the product measure $mu = mu_Dtimes mu_E$ as $beta to +infty$. We also supply conditions under which $mu_infty$ can be obtained as the athermal limit of carefully thermalized $(mu_{h,beta_h})$ sequences of empirical data sets $(mu_h)$ approximating weakly an unknown likelihood function $mu$. In particular, we find that the cooling sequence $beta_h to +infty$ must be slow enough, corresponding to quenching, in order for the proper limit $mu_infty$ to be delivered. Finally, we derive explicit analytic expressions for expectations $mathbb{E}[f]$ of outcomes $f$ that are explicit in the data, thus demonstrating the feasibility of the model-free data-driven paradigm as regards making convergent inferences directly from the data without recourse to intermediate modeling steps.